Question 1

Find the integral using an appropriate trigonometric substitution. $\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 36 } } d x$

Accepted Answer

A)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 36 \right) ^ { 3 / 2 } \sqrt { x ^ { 2 } + 36 } + C$ 
B)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 72 \right) \sqrt { x ^ { 2 } + 36 } + C$ 
C)  $\frac { 1 } { 3 } \left( x ^ { 2 } - 72 \right) \sqrt { x ^ { 2 } + 36 } + C$ 
D)  $\frac { 1 } { 3 } \left( x ^ { 2 } - 36 \right) ^ { 3 / 2 } \sqrt { x ^ { 2 } + 36 } + C$ 
A)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 36 \right) ^ { 3 / 2 } \sqrt { x ^ { 2 } + 36 } + C$ 
B)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 72 \right) \sqrt { x ^ { 2 } + 36 } + C$ 
C)  $\frac { 1 } { 3 } \left( x ^ { 2 } - 72 \right) \sqrt { x ^ { 2 } + 36 } + C$ 
D)  $\frac { 1 } { 3 } \left( x ^ { 2 } - 36 \right) ^ { 3 / 2 } \sqrt { x ^ { 2 } + 36 } + C$

Question 2

You are given function $f$ defined on an interval $[ a , b ]$, the number $n$ of subintervals of equal length $\Delta x = \frac { ( b - a ) } { n }$, and the evaluation points $c _ { k }$ in $\left[ x _ { k - 1 } , x _ { k } \right]$.
(a) Sketch the graph of $f$ and the rectangles associated with the Riemann sum for $f$ on $[ a , b ]$, and
(b) find the Riemann sum.
$f ( x ) = 2 x - 3 , \quad [ 0,2 ] , \quad n = 4 , \quad c _ { k }$ is the right endpoint

Accepted Answer

The answer of You are given function $f$ defined on...

Question 3

Evaluate the integral if it exists. $$\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x$$

Accepted Answer

A)  $1 - 10 \ln x + C$ 
B) 
$x - 10 \ln x - \frac { 25 } { x } + C$ 
C)  $\ln x - 25 x + C$ 
D) 
$x - \frac { 1 } { 10 \ln x } + C$ 
E)  none of these 
A)  $1 - 10 \ln x + C$ 
B) 
$x - 10 \ln x - \frac { 25 } { x } + C$ 
C)  $\ln x - 25 x + C$ 
D) 
$x - \frac { 1 } { 10 \ln x } + C$ 
E)  none of these

Question 4

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
$$h ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { 5 z ^ { 2 } } { z ^ { 4 } + 1 } d z$$

Accepted Answer

A)  $\frac { 5 \sqrt { x } } { x ^ { 2 } + 1 }$ 
B)  $\frac { \sqrt { x ^ { 2 } + 1 } } { 5 }$ 
C) 
$$\frac { 5 \sqrt { x } } { 2 \left( x ^ { 2 } + 1 \right) }$$ 
D)  $\frac { \sqrt { x + 5 } } { x ^ { 2 } + 5 }$ 
E)  none of these 
A)  $\frac { 5 \sqrt { x } } { x ^ { 2 } + 1 }$ 
B)  $\frac { \sqrt { x ^ { 2 } + 1 } } { 5 }$ 
C) 
$$\frac { 5 \sqrt { x } } { 2 \left( x ^ { 2 } + 1 \right) }$$ 
D)  $\frac { \sqrt { x + 5 } } { x ^ { 2 } + 5 }$ 
E)  none of these

Question 5

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $$h ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { z ^ { 2 } } { z ^ { 4 } + 1 } d z$$

Accepted Answer

A)  $\frac { \sqrt { x + 1 } } { x ^ { 2 } + 2 }$ 
B)  $\frac { \sqrt { x ^ { 2 } + 1 } } { 2 }$ 
C) 
$\frac { \sqrt { x } } { 2 \left( x ^ { 2 } + 1 \right) }$ 
D)  $\frac { \sqrt { x } } { x ^ { 2 } + 1 }$ 
E)  none of these 
A)  $\frac { \sqrt { x + 1 } } { x ^ { 2 } + 2 }$ 
B)  $\frac { \sqrt { x ^ { 2 } + 1 } } { 2 }$ 
C) 
$\frac { \sqrt { x } } { 2 \left( x ^ { 2 } + 1 \right) }$ 
D)  $\frac { \sqrt { x } } { x ^ { 2 } + 1 }$ 
E)  none of these

Question 6

$$\text { Evaluate } \int _ { 0 } ^ { 3 } \left( x + \sqrt { 9 - x ^ { 2 } } \right) d x \text { by interpreting it in terms of areas. }$$

Accepted Answer

@#LAT-DLM&
@#LAT-DLM& Area of

Question 7

Evaluate the integral $\int _ { 0 } ^ { z / 20 } \frac { e ^ { \tan 5 x } } { \cos ^ { 2 } 5 x } d x$.

Accepted Answer

A)  $5 ( e - 1 )$ 
B)  $20 \left( e ^ { 5 } - 1 \right)$ 
C)  $\frac { e - 1 } { 5 }$ 
D)  $\frac { e ^ { 5 } - 1 } { 20 }$ 
A)  $5 ( e - 1 )$ 
B)  $20 \left( e ^ { 5 } - 1 \right)$ 
C)  $\frac { e - 1 } { 5 }$ 
D)  $\frac { e ^ { 5 } - 1 } { 20 }$

Question 8

A table of values of an increasing function $f ( x )$ is shown. Use the table to find an upper estimate of:
$\begin{array}{|c|c|}
\hline x & f(x) \
\hline 0 & -46 \
\hline 5 & -30 \
\hline 10 & -20 \
\hline 15 & 4 \
\hline 20 & 15 \
\hline 25 & 20 \
\hline
\end{array}$

Accepted Answer

The answer of A table of values of an increasing...

Question 9

Evaluate the definite integral.
$$\int _ { 0 } ^ { x / 6 } \sin 8 t d t$$

Accepted Answer

A)  $0.188$ 
B)  $3.25$ 
C)  $- 0.045$ 
D)  $0.35$ 
E)  $2.891$ 
A)  $0.188$ 
B)  $3.25$ 
C)  $- 0.045$ 
D)  $0.35$ 
E)  $2.891$

Question 10

The velocity graph of a car accelerating from rest to a speed of $7 \mathrm {~km} / \mathrm { h }$ over a period of 10 seconds is shown. Estimate to the nearest integer the distance traveled during this period. Use a right sum with $n = 10$.

Accepted Answer

The answer of The velocity graph of a car accelerating...

Question 11

Find the indefinite integral. $$\int \frac { \cot ^ { 2 } x } { \cos ^ { 2 } x } d x$$

Accepted Answer

A)  $\cos ^ { 2 } x + C$ 
B)  $\tan x + C$ 
C)  $\csc ^ { 2 } x + C$ 
D)  $- \cot x + C$ 
A)  $\cos ^ { 2 } x + C$ 
B)  $\tan x + C$ 
C)  $\csc ^ { 2 } x + C$ 
D)  $- \cot x + C$

Question 12

Evaluate the indefinite integral. $$\int \sec ^ { 2 } x \tan x d x$$

Accepted Answer

The answer of Evaluate the indefinite integral. \[\int \sec ^...

Question 13

Evaluate the indefinite integral.
$\int 7 e ^ { \cos x } \sin x d x$

Accepted Answer

A)  $- e ^ { \cos x } \sin x + C$ 
B)  $- 7 e ^ { \cos x } + C$ 
C)  $e ^ { 7 \sin x } + C$ 
D)  $7 e ^ { \cos x } \sin x + C$ 
E)  $- 7 \sin \left( e ^ { \cos x } \right) + C$ 
A)  $- e ^ { \cos x } \sin x + C$ 
B)  $- 7 e ^ { \cos x } + C$ 
C)  $e ^ { 7 \sin x } + C$ 
D)  $7 e ^ { \cos x } \sin x + C$ 
E)  $- 7 \sin \left( e ^ { \cos x } \right) + C$

Question 14

Let $F ( x ) = \int _ { 5 } ^ { x } 5 t ^ { 2 } d t$.
a. Use Part 1 of the Fundamental Theorem of Calculus to find $F ^ { \prime } ( x )$.
b. Use Part 2 of the Fundamental Theorem of Calculus to integrate $\int _ { 5 } ^ { x } 5 t ^ { 2 } d t$ to obtain an alternative expression for $F ( x )$.
c. Differentiate the expression for $F ( x )$ found in part (b). The Fundamental Theorem of Calculus, Part 1 If $f$ is continuous on $[ a , b ]$, then the function $F$ defined by $F ( x ) = \int _ { a } ^ { x } f ( t ) d t \quad a \leq x \leq b$ is differentiable on $( a , b )$, and $F ^ { \prime } ( x ) = \frac { d } { d x } \int _ { a } ^ { x } f ( t ) d t = f ( x )$ The Fundamental Theorem of Calculus, Part 2
If $f$ is continuous on $[ a , b ]$, then $\int _ { a } ^ { b } f ( x ) d x = F ( b ) - F ( a )$ where $F$ is any antiderivative of $f$, that is, $F ^ { t } = f$.

Accepted Answer

The answer of Let \(F ( x ) = \int...

Question 15

If $\int _ { 0 } ^ { 6 } f ( x ) d x = 13$ and $\int _ { 0 } ^ { 4 } f ( x ) d x = 7$, find $\int _ { 4 } ^ { 6 } f ( x ) d x$

Accepted Answer

A)  7 
B) -13 
C)  20 
D)  6 
E)  13 
A)  7 
B) -13 
C)  20 
D)  6 
E)  13

Question 16

Evaluate the integral.
$\int _ { 0 } ^ { 6 } \frac { 1 } { 36 + t ^ { 2 } } d t$

Accepted Answer

A)  $\frac { 4 \pi } { 5 }$ 
B)  $\frac { \pi } { 10 }$ 
C)  $\frac { \pi } { 24 }$ 
D)  $\frac { \pi } { 2 }$ 
E)  $10 \pi$ 
A)  $\frac { 4 \pi } { 5 }$ 
B)  $\frac { \pi } { 10 }$ 
C)  $\frac { \pi } { 24 }$ 
D)  $\frac { \pi } { 2 }$ 
E)  $10 \pi$

Question 17

Express the integral as a limit of sums.Then evaluate the limit. $$\int _ { 0 } ^ { x } \sin 3 x d x$$

Accepted Answer

The answer of Express the integral as a limit of...

Question 18

The acceleration function of a body moving along a coordinate line is $a ( t ) = - 7 \cos 2 t - 8 \sin 2 t \quad t \geq 0$ Find its velocity and position functions at any time $t$ if it is located at the origin and has an initial velocity of $4 \mathrm {~m} / \mathrm { sec }$.

Accepted Answer

The answer of The acceleration function of a body moving...

Question 19

Express the limit as a definite integral on the given interval. $$\lim _ { n \rightarrow \infty } \sum _ { i = 1 } ^ { n } \left( 6 r _ { i } \sin r _ { i } \right) \Delta r , [ 6,12 ]$$

Accepted Answer

The answer of Express the limit as a definite integral...

Question 20

The marginal cost of manufacturing x yards of a certain fabric is $$C ^ {t } ( x ) = 3 - 0.01 x + 0.000006 x ^ { 2 }$$
in dollars per yard. Find the increase in cost if the production level is raised from 1500 yards to 5500 yards.

Accepted Answer

A)  $\$ 188,000.00$ 
B)  $\$ 198,000.00$ 
C)  $\$ 178,000.00$ 
D)  $\$ 218,000.00$ 
E)  $\$ 208,000.00$ 
A)  $\$ 188,000.00$ 
B)  $\$ 198,000.00$ 
C)  $\$ 178,000.00$ 
D)  $\$ 218,000.00$ 
E)  $\$ 208,000.00$

Find the integral using an appropriate trigonometric substitution. $\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 36 } } d x$

Evaluate the integral if it exists. $\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x$

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $h ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { 5 z ^ { 2 } } { z ^ { 4 } + 1 } d z$

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $h ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { z ^ { 2 } } { z ^ { 4 } + 1 } d z$

$\text { Evaluate } \int _ { 0 } ^ { 3 } \left( x + \sqrt { 9 - x ^ { 2 } } \right) d x \text { by interpreting it in terms of areas. }$

Evaluate the integral $\int _ { 0 } ^ { z / 20 } \frac { e ^ { \tan 5 x } } { \cos ^ { 2 } 5 x } d x$ .

A table of values of an increasing function $f ( x )$ is shown. Use the table to find an upper estimate of: x f(x) 0 -46 5 -30 10 -20 15 4 20 15 25 20

Evaluate the definite integral. $\int _ { 0 } ^ { x / 6 } \sin 8 t d t$

Find the indefinite integral. $\int \frac { \cot ^ { 2 } x } { \cos ^ { 2 } x } d x$

Evaluate the indefinite integral. $\int \sec ^ { 2 } x \tan x d x$

Evaluate the indefinite integral. $\int 7 e ^ { \cos x } \sin x d x$

If $\int _ { 0 } ^ { 6 } f ( x ) d x = 13$ and $\int _ { 0 } ^ { 4 } f ( x ) d x = 7$ , find $\int _ { 4 } ^ { 6 } f ( x ) d x$

Evaluate the integral. $\int _ { 0 } ^ { 6 } \frac { 1 } { 36 + t ^ { 2 } } d t$

Express the integral as a limit of sums.Then evaluate the limit. $\int _ { 0 } ^ { x } \sin 3 x d x$

The acceleration function of a body moving along a coordinate line is $a ( t ) = - 7 \cos 2 t - 8 \sin 2 t \quad t \geq 0$ Find its velocity and position functions at any time $t$ if it is located at the origin and has an initial velocity of $4 \mathrm {~m} / \mathrm { sec }$ .

Express the limit as a definite integral on the given interval. $\lim _ { n \rightarrow \infty } \sum _ { i = 1 } ^ { n } \left( 6 r _ { i } \sin r _ { i } \right) \Delta r , [ 6,12 ]$

The marginal cost of manufacturing x yards of a certain fabric is $C ^ {t } ( x ) = 3 - 0.01 x + 0.000006 x ^ { 2 }$ in dollars per yard. Find the increase in cost if the production level is raised from 1500 yards to 5500 yards.

Functions and Models

Limits and Derivatives

Differentiation Rules

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

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Parametric Equations and Polar Coordinates

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Exam 4: Applications of Differentiation

Find the integral using an appropriate trigonometric substitution. ∫x3x2+36dx\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 36 } } d x∫x2+36​x3​dx

Evaluate the integral if it exists. ∫(5−xx)2dx\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x∫(x5−x​)2dx

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(x)=∫1x5z2z4+1dzh ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { 5 z ^ { 2 } } { z ^ { 4 } + 1 } d zh(x)=∫1x​​z4+15z2​dz

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(x)=∫1xz2z4+1dzh ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { z ^ { 2 } } { z ^ { 4 } + 1 } d zh(x)=∫1x​​z4+1z2​dz

Evaluate ∫03(x+9−x2)dx by interpreting it in terms of areas. \text { Evaluate } \int _ { 0 } ^ { 3 } \left( x + \sqrt { 9 - x ^ { 2 } } \right) d x \text { by interpreting it in terms of areas. } Evaluate ∫03​(x+9−x2​)dx by interpreting it in terms of areas.

Evaluate the integral ∫0z/20etan⁡5xcos⁡25xdx\int _ { 0 } ^ { z / 20 } \frac { e ^ { \tan 5 x } } { \cos ^ { 2 } 5 x } d x∫0z/20​cos25xetan5x​dx .

A table of values of an increasing function f(x)f ( x )f(x) is shown. Use the table to find an upper estimate of: x f(x) 0 -46 5 -30 10 -20 15 4 20 15 25 20

Evaluate the definite integral. ∫0x/6sin⁡8tdt\int _ { 0 } ^ { x / 6 } \sin 8 t d t∫0x/6​sin8tdt

The velocity graph of a car accelerating from rest to a speed of 7 km/h7 \mathrm {~km} / \mathrm { h }7 km/h over a period of 10 seconds is shown. Estimate to the nearest integer the distance traveled during this period. Use a right sum with n=10n = 10n=10 .

Find the indefinite integral. ∫cot⁡2xcos⁡2xdx\int \frac { \cot ^ { 2 } x } { \cos ^ { 2 } x } d x∫cos2xcot2x​dx

Evaluate the indefinite integral. ∫sec⁡2xtan⁡xdx\int \sec ^ { 2 } x \tan x d x∫sec2xtanxdx

Evaluate the indefinite integral. ∫7ecos⁡xsin⁡xdx\int 7 e ^ { \cos x } \sin x d x∫7ecosxsinxdx

If ∫06f(x)dx=13\int _ { 0 } ^ { 6 } f ( x ) d x = 13∫06​f(x)dx=13 and ∫04f(x)dx=7\int _ { 0 } ^ { 4 } f ( x ) d x = 7∫04​f(x)dx=7 , find ∫46f(x)dx\int _ { 4 } ^ { 6 } f ( x ) d x∫46​f(x)dx

Evaluate the integral. ∫06136+t2dt\int _ { 0 } ^ { 6 } \frac { 1 } { 36 + t ^ { 2 } } d t∫06​36+t21​dt

Express the integral as a limit of sums.Then evaluate the limit. ∫0xsin⁡3xdx\int _ { 0 } ^ { x } \sin 3 x d x∫0x​sin3xdx

Express the limit as a definite integral on the given interval. lim⁡n→∞∑i=1n(6risin⁡ri)Δr,[6,12]\lim _ { n \rightarrow \infty } \sum _ { i = 1 } ^ { n } \left( 6 r _ { i } \sin r _ { i } \right) \Delta r , [ 6,12 ]limn→∞​∑i=1n​(6ri​sinri​)Δr,[6,12]

The marginal cost of manufacturing x yards of a certain fabric is Ct(x)=3−0.01x+0.000006x2C ^ {t } ( x ) = 3 - 0.01 x + 0.000006 x ^ { 2 }Ct(x)=3−0.01x+0.000006x2 in dollars per yard. Find the increase in cost if the production level is raised from 1500 yards to 5500 yards.

Functions and Models

Limits and Derivatives

Differentiation Rules

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

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Find the integral using an appropriate trigonometric substitution. $\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 36 } } d x$

Evaluate the integral if it exists. $\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x$

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $h ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { 5 z ^ { 2 } } { z ^ { 4 } + 1 } d z$

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $h ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { z ^ { 2 } } { z ^ { 4 } + 1 } d z$

$\text { Evaluate } \int _ { 0 } ^ { 3 } \left( x + \sqrt { 9 - x ^ { 2 } } \right) d x \text { by interpreting it in terms of areas. }$

Evaluate the integral $\int _ { 0 } ^ { z / 20 } \frac { e ^ { \tan 5 x } } { \cos ^ { 2 } 5 x } d x$ .

A table of values of an increasing function $f ( x )$ is shown. Use the table to find an upper estimate of: x f(x) 0 -46 5 -30 10 -20 15 4 20 15 25 20

Evaluate the definite integral. $\int _ { 0 } ^ { x / 6 } \sin 8 t d t$

Find the indefinite integral. $\int \frac { \cot ^ { 2 } x } { \cos ^ { 2 } x } d x$

Evaluate the indefinite integral. $\int \sec ^ { 2 } x \tan x d x$

Evaluate the indefinite integral. $\int 7 e ^ { \cos x } \sin x d x$

If $\int _ { 0 } ^ { 6 } f ( x ) d x = 13$ and $\int _ { 0 } ^ { 4 } f ( x ) d x = 7$ , find $\int _ { 4 } ^ { 6 } f ( x ) d x$

Evaluate the integral. $\int _ { 0 } ^ { 6 } \frac { 1 } { 36 + t ^ { 2 } } d t$

Express the integral as a limit of sums.Then evaluate the limit. $\int _ { 0 } ^ { x } \sin 3 x d x$

Express the limit as a definite integral on the given interval. $\lim _ { n \rightarrow \infty } \sum _ { i = 1 } ^ { n } \left( 6 r _ { i } \sin r _ { i } \right) \Delta r , [ 6,12 ]$

The marginal cost of manufacturing x yards of a certain fabric is $C ^ {t } ( x ) = 3 - 0.01 x + 0.000006 x ^ { 2 }$ in dollars per yard. Find the increase in cost if the production level is raised from 1500 yards to 5500 yards.